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A379203 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 3)^(n+1).

%I A379203 #8 Dec 20 2024 23:38:47
%S A379203 1,5,34,290,2820,29629,327301,3744868,43981858,527126689,6420981368,
%T A379203 79260797860,989306411413,12464737320229,158320378037652,
%U A379203 2025016002188169,26060398562711196,337197048402240367,4384067953773647268,57245716462267462224,750403639664344374239,9871281245683966836462
%N A379203 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 3)^(n+1).
%C A379203 Related identity: Sum_{n=-oo..+oo} x^n*(y - x^n)^n = 0, which holds formally for all y.
%H A379203 Paul D. Hanna, <a href="/A379203/b379203.txt">Table of n, a(n) for n = 1..300</a>
%F A379203 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A379203 (1) 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 3)^(n+1).
%F A379203 (2) 1/x = Sum_{n=-oo..+oo} A(x)^(2*n) * (A(x)^n - 3)^n.
%F A379203 (3) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 + 3*A(x)^(n+1))^n.
%F A379203 (4) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 - 3*A(x)^(n+1))^(n+1).
%F A379203 (5) A(B(x)) = x where B(x) = 1/( Sum_{n=-oo..+oo} x^n * (x^n + 3)^(n+1) ).
%e A379203 G.f.: A(x) = x + 5*x^2 + 34*x^3 + 290*x^4 + 2820*x^5 + 29629*x^6 + 327301*x^7 + 3744868*x^8 + 43981858*x^9 + 527126689*x^10 + ...
%e A379203 SPECIFIC VALUES.
%e A379203 A(t) = 1/7 at t = 0.069769772400266469707360138034033927488705716660080...
%e A379203   where t = 1/Sum_{n=-oo..+oo} (1 + 3*7^(n-1))^n / 7^(n^2-1).
%e A379203 A(t) = 1/8 at t = 0.067295105779482404156544832668824160420208234924667...
%e A379203   where t = 1/Sum_{n=-oo..+oo} (1 + 3*8^(n-1))^n / 8^(n^2-1).
%e A379203 A(t) = 1/9 at t = 0.064327556053208007320009998534415581932268509899202...
%e A379203   where t = 1/Sum_{n=-oo..+oo} (1 + 3*9^(n-1))^n / 9^(n^2-1).
%e A379203 A(t) = 1/10 at t = 0.06126924119589872239866986020862532219839002819792...
%e A379203   where t = 1/Sum_{n=-oo..+oo} (1 + 3*10^(n-1))^n / 10^(n^2-1).
%e A379203 A(1/15) = 0.12166176397390884847529063617720403039492284665035...
%e A379203 A(1/16) = 0.10420546336336096378642246758350885785023968035181...
%e A379203 A(1/20) = 0.07053009254165709187694647754531300907207762301254...
%o A379203 (PARI) {a(n) = my(V=[0, 1], A); for(i=1, n, V=concat(V, 0); A = Ser(V);
%o A379203 V[#V] = polcoef( sum(m=-#A, #A, A^m*(A^m + 3)^(m+1) ), #V-3); ); polcoef(A, n)}
%o A379203 for(n=1, 40, print1(a(n), ", "))
%Y A379203 Cf. A379200, A379199, A166952, A378264, A379202, A379204, A379205.
%K A379203 nonn
%O A379203 1,2
%A A379203 _Paul D. Hanna_, Dec 20 2024